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Properties

Label 2-35-35.9-c1-0-1
Degree $2$
Conductor $35$
Sign $0.830 + 0.556i$
Analytic cond. $0.279476$
Root an. cond. $0.528655$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (−0.500 − 0.866i)4-s + (1.23 + 1.86i)5-s − 0.999·6-s + (−2.59 − 0.5i)7-s + 3i·8-s + (−1 + 1.73i)9-s + (−0.133 − 2.23i)10-s + (−0.866 − 0.499i)12-s − 2i·13-s + (2 + 1.73i)14-s + (2 + i)15-s + (0.500 − 0.866i)16-s + (1.73 − i)17-s + (1.73 − i)18-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (−0.250 − 0.433i)4-s + (0.550 + 0.834i)5-s − 0.408·6-s + (−0.981 − 0.188i)7-s + 1.06i·8-s + (−0.333 + 0.577i)9-s + (−0.0423 − 0.705i)10-s + (−0.249 − 0.144i)12-s − 0.554i·13-s + (0.534 + 0.462i)14-s + (0.516 + 0.258i)15-s + (0.125 − 0.216i)16-s + (0.420 − 0.242i)17-s + (0.408 − 0.235i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.830 + 0.556i$
Analytic conductor: \(0.279476\)
Root analytic conductor: \(0.528655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1/2),\ 0.830 + 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.579207 - 0.176127i\)
\(L(\frac12)\) \(\approx\) \(0.579207 - 0.176127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.23 - 1.86i)T \)
7 \( 1 + (2.59 + 0.5i)T \)
good2 \( 1 + (0.866 + 0.5i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.866 + 0.5i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.33 - 2.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (-5.19 + 3i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11iT - 83T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.75630795189748001635313477401, −15.13336516674055792988612381600, −13.99687654344275496005780677264, −13.23183529941576512737781814143, −11.19194163352309747159901154511, −10.12022840969341865944050524293, −9.152782447120841016587155346774, −7.47778755486552906944461559803, −5.73394084151602609198180591633, −2.70491994321594748957871473769, 3.70880498029801417858296829336, 6.08736534843365498823103052728, 7.965175190318869538144019382111, 9.301335416853110947527723441366, 9.663022480577508129781761027413, 12.16344503206688543355672839558, 13.08828761368256559470201926028, 14.38870768282604730026847817490, 16.01776057877680642951370267214, 16.59599889359784038047228436459

Graph of the $Z$-function along the critical line